- Research Article
- Open Access

# Multiplicity of Positive and Nodal Solutions for Nonhomogeneous Elliptic Problems in Unbounded Cylinder Domains

- Tsing-San Hsu
^{1}Email author

**Received:**13 March 2009**Accepted:**7 May 2009**Published:**9 June 2009

## Abstract

We show that if and satisfy some suitable conditions, then the Dirichlet problem in has a solution that changes sign in , in addition to two positive solutions where is an unbounded cylinder domain in .

## Keywords

- Positive Constant
- Minimization Problem
- Variational Principle
- Multiplicity Result
- Ground State Solution

## 1. Introduction

where is a bounded smooth domain, is a smooth unbounded cylinder domain in .

It is assumed that and satisfy the following assumptions:

(*f* 1)
;

where is the first positive eigenvalue of the Dirichlet problem in .

For the homogeneous case, that is, , Zhu [1] has established the existence of a positive solution and a nodal solution of problem (1.2) in provided satisfies in and as for some positive constants and . More recently, Hsu [2] extended the results of Zhu [1] with to an unbounded cylinder . Let us recall that, by a nodal solution we mean the solution of problem (1.2) with change of sign.

For the nonhomogeneous case ( ),Adachi and Tanaka [3] have showed that problem (1.2) has at least four positive solutions in for and satisfy some suitable conditions, but we place particular emphasis on the existence of nodal solutions. More recently, Chen [4] considered the multiplicity results of both positive and nodal solutions of problem (1.2) in . She has showed that problem (1.2) has at least two positive solutions and one nodal solution in when and satisfy some suitable assumptions.

In the present paper, motivated by [4] we extend and improve the paper by Chen [4]. We will deal with unbounded cylinder domains instead of the entire space and also obtain the same results as in [4]. Our arguments are similar to those in [5, 6], which are based on Ekeland's variational principle [7].

Now, we state our main results.

Theorem 1.1.

Assume hold and satisfies assumption .

Then problem (1.2) has at least two positive solutions and in . Furthermore, and satisfy , and is a local minimizer of where is the energy functional of problem (1.2).

Theorem 1.2.

Assume hold and satisfies assumption .

Then problem (1.2) has a nodal solution in in addition to two positive solutions and .

For the case , we also have obtained the same results as in Theorems 1.1 and 1.2.

Theorem 1.3.

Assume hold and satisfies assumption .

Then problem (1.2) has at least two positive solutions and in . Furthermore, and satisfy , and is a local minimizer of where is the energy functional of problem (1.2).

Theorem 1.4.

Assume hold and satisfies assumption below.

Then problem (1.2) has a nodal solution in in addition to two positive solutions and .

where , , and is a bounded domain in . They found that (1.10) possesses infinitely many solutions. More recently, Tarantello [5] proved that if is the critical Sobolev exponent and satisfying suitable conditions, then (1.10) admits two solutions. For the case when is an unbounded domain, Cao and Zhou [10], Cîrstea and Rădulescu [11], and Ghergu and Rădulescu [12] have been investigated the analogue equation (1.10) involving a subcritical exponent in . Furthermore, Rădulescu and Smets [13] proved existence results for nonautonomous perturbations of critical singular elliptic boundary value problems on infinite cones.

This paper is organized as follows. In Section 2, we give some notations and preliminary results. In Section 3, we will prove Theorem 1.1. In Section 4, we establish the existence of nodal solutions.

## 2. Preliminaries

In this paper, we always assume that is an unbounded cylinder domain or . Let for , and let be the first positive eigenfunction of the Dirichlet problem in with eigenvalue unless otherwise specified. We denote by and ( ) universal constants, maybe the constants here should be allowed to depend on and , unless some statement is given. Now we begin our discussion by giving some definitions and some known results.

here and from now on, we omit " " and " " in all the integration if there is no other indication. It is well known that is of in and the solutions of problem (1.2) are the critical points of the energy functional (see Rabinowitz [14]).

Easy computation shows that is bounded from below in the set . Note that contains every nonzero solution of (1.2).

Let us introduce the problem at infinity associated with problem (1.2) as

Furthermore, from Hsu [2] we can deduce that for any there exist positive constants such that, for all ,

We also quote the following lemma (see Hsu [17] or K. -J. Chen et al. [18] for the proof) about the decay of positive solution of problem (1.2) which we will use later.

Lemma 2.1.

Assume and hold. If is a positive solution of problem (1.2), then

(i) for all ;

(ii) as uniformly for and for any ;

We end this preliminaries by the following definition.

Definition 2.2.

Let , be a Banach space and .

- (ii)
We say that satisfies the condition if any -sequence in for has a convergent subsequence.

## 3. Proof of Theorem 1.1

In this section, we will establish the existence of two positive solutions of problem (1.2).

First, we quote some lemmas for later use (see the proof of Tarantello [5] or Chen [4, Lemmas 2.2, 2.3, and 2.4]).

Lemma 3.1.

and .

Lemma 3.2.

Lemma 3.3.

Apply Lemmas 3.1, 3.2, 3.3, and Ekeland variational principle [7], and we can establish the existence of the first positive solution.

Proposition 3.4.

Assume and hold, then the minimization problem is achieved at a point which is a critical point for . Moreover, if and , then is a positive solution of problem (1.2) and is a local minimizer of .

Proof.

Modifying the proof of Chen [4, Proposition 2.5]. Here we omit it.

We will establish the existence of the second positive solution of problem (1.2) by proving that satisfies the -condition.

Proposition 3.5.

Assume and hold, then satisfies the -condition with .

Proof.

which is a contradiction. Therefore, and we conclude that strongly in .

Let , let , and let be a constant, we denote and for where is the ground state solution of problem (2.6) and is the first positive solution of problem (1.2).

Proposition 3.6.

The following estimates are important to find a path which lies below the first level of the break down of the condition. Here we use an interaction phenomenon between and .

To give a proof of Proposition 3.6, we need to establish some lemmas.

Lemma 3.7.

Proof.

Lemma 3.8.

Proof.

Since as the lemma follows from the Lebesgue's dominated convergence theorem.

Now, we give the proof of Proposition 3.6.

The Proof of Proposition 3.6

Thus from (3.26) and (3.32)–(3.35), we obtain (3.13). This completes the proof of Proposition 3.6.

Proposition 3.9.

For , there exists a -sequence for . In particular, we have .

Proof.

and .

We will prove that there exists such that . Denote . Since , we have

hence .

disconnects in exactly two components, so we can find an such that . Therefore , which follows from Proposition 3.6.

Analogously to the proof of Proposition 3.4, by the Ekeland variational principle we can show that there exists a -sequence for .

Proposition 3.10.

Assume and hold, then the functional has a minimizer which is also a critical point of and for .

Proof.

From Propsitions 3.5 and 3.9, we can deduce that strongly in . Consequently, is a critical point of , (since is closed) and .

By Lemma 3.1, we can choose a number such that . Since . Applying Lemma 3.1 again, we conclude that

Hence . So we can always take . By the maximum principle for weak solutions (see Gilbarg and Trudinger [20]) we can show that if , then in .

The proof of Theorem 1.1

By Propositions 3.4 and 3.10, we obtain the conclusion of Theorem 1.1.

## 4. Existence of Nodal Solution

Then we have

- (a)
If , then the minimization problem (4.2) attains its infimum at a point which defines a sign changing critical point of .

- (b)
Analogously, if the same conclusion holds for the minimization problem (4.3).

Proof.

The proof is almost the same as that in Tarantello [6, Proposition 3.1] .

It is clear that . Since satisfies condition only locally, we need the following upper bound for . Recall that , and where and is the ground state solution of problem (2.6).

Lemma 4.2.

Proof.

Therefore, by the continuity of , we can find such that . This gives (4.8) with and .

To prove (4.7), we only need to estimate for and . First, it is obvious that the structure of guarantees the existence of (independent of large) such that , for all . On the other hand, for , since is continuous in , there exists small enough such that

Without loss of generality, we may assume , and where , and are given in and , respectively.

This completes the proof of Lemma 4.2.

Proposition 4.3.

Assume and hold. If and , then the minimization problem attains its infimum at which defines a changing sign critical point of .

Proof.

It is obvious that is closed. Exactly as in the proof of [6, Proposition 3.2], by means of Ekeland's principle, we derive a -sequence for . In particular, we have , for some constants and . Thus, we can take a subsequence, also denoted by , such that weakly in . We start by showing that .

Indeed, if by contradiction we assume, for instant, that , then we can deduce that

However, by Lemma 4.2, ; that is, which contradicts (4.27). A similar argument applies to . Therefore, is a weak solution of problem (1.2) changing sign and .

Set and with weakly in . Note that

We claim that . Indeed, we assume is bounded below, as above, (4.28) and (4.32) imply , contradicting (4.31). In the same way, if , we can also prove . Hence we have or ; that is, or . By assumptions and , we conclude that .

If we write with weakly in , we have

We claim that . Indeed, we assume is bounded below, as above, (4.33) imply , contradicting (4.34). Consequently, strongly in and .

The Proof of Theorems 1.2–1.4

The conclusion of Theorem 1.2 follows immediately from Theorem 1.2 and Propositions 4.1 and 4.3. With the same argument, we also have that Theorems 1.3 and 1.4 hold for .

## Authors’ Affiliations

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